Exchange of differentiation and limit on a closed interval

Question

For a differentiable function $f(x,y)$, is the following true,$$\lim_{x\to 0}\frac{\partial f(x,y)}{\partial y} = \frac{\partial \lim_{x\to 0} f(x,y)}{\partial y}$$ on a closed interval $y\in[0,\bar{y}]$? If not, when is it true? Similar question seems to have been asked, e.g.,Interchange of partial derivative and limit. But no answer has been provided.